Space-time encoding in wireless communication systems

ABSTRACT

A disclosed transmitter for wireless communication includes multiple transmitting antennas, a symbol mapper for mapping an input block including multiple binary bits and representing information to be transmitted to a symbol representing an ordered plurality of complex numbers, a space-time encoder for applying an encoding operator to the symbol to produce a vectorized space-time codeword defining electrical signals to be transmitted by the transmitter, the encoding operator being dependent on a set of predefined stabilizer generators, and circuitry to collectively transmit, by the antennas to multiple receiving antennas of a receiver over a wireless transmission channel, the electrical signals defined by the vectorized space-time codeword. The receiver includes a space-time decoder for recovering the symbol from the electrical signals transmitted by the transmitter using a decoding operation that is based on maximum likelihood inference, and a symbol de-mapper for recovering the input block from the symbol.

RELATED APPLICATIONS

This application claims priority to U.S. Provisional Application No.62/948,974 filed Dec. 17, 2019, entitled “SPACE-TIME ENCODING INWIRELESS COMMUNICATION SYSTEMS,” which is incorporated herein byreference in its entirety.

STATEMENT OF GOVERNMENT INTEREST

This invention was made with government support under Grant no.N00014-17-1-2107 awarded by the Office of Naval Research. The governmenthas certain rights in the invention.

FIELD OF THE DISCLOSURE

The present disclosure relates generally to the field of wirelesscommunication systems and more particularly, to systems and methods forspace-time encoding of information to be transmitted over a wirelesscommunication channel.

DESCRIPTION OF THE RELATED ART

The fifth-generation (5G) cellular standard, like its predecessors,promises significant improvements to broadband data rates. The fullobjectives of 5G, however, also demand support for a variety of novelcommunication scenarios not directly pertinent to the enhancement ofmobile broadband. Specifically, ultra-reliable low-latency communication(URLLC) is a 5G effort aimed at providing support for “mission-critical”communication, characterized by requirements for low throughputcommunication with especially small error rates (e.g., 10⁻⁹ block errorprobability) and short latencies (e.g., 1 millisecond end-to-end).Similarly, massive machine-to-machine (MM2M) type communication is a 5Geffort entailing support for highly dense networks of wireless devicescommunicating at low-to-moderate data rates. Developing reliablecommunication in these paradigms remains crucial to the continueddevelopment of emerging industries such as autonomous vehicles andindustrial automation networks.

URLLC and MM2M system design efforts demand especially stringentconstraints on the packet size and the block error probability, inparticular. Code design in this regime has been aided by the developmentof information-theoretic tools analyzing communication at finiteblocklengths. Such tools provide insight into the ultimate performancelimits of communication networks operating outside the “asymptoticregion” of blocklength in a manner that asymptotic information-theoreticanalyses could not.

In a departure from communication system designs with very longblocklengths, which overwhelmingly favor spatial multiplexing approachesas a means of improving spectral efficiency, analyses inblocklength-constrained settings reveal a significant rate penaltyarising from a limited blocklength. Spatial degrees of freedom in theform of multiple antennas, therefore, are no longer necessary in orderto operate near the channel capacity but can still be directed towardreducing the block error probability. This invites a reconsideration ofthe diversity-multiplexing tradeoff in the constrained multiantennasetting, which has been previously characterized in the high-SNRsetting.

Noncoherent communication may be characterized by a lack of channelknowledge at both the transmitter and the receiver, or by the fact thatit does not require an explicit estimate of the propagation channel atthe receiver. This scenario is assumed when the propagation channelchanges rapidly, or with frequency hopping waveforms, where the trainingrequired for coherent communication takes too much overhead to bepractical.

SUMMARY

This disclosure relates to systems, methods, and apparatus forspace-time encoding in wireless communication systems. In one aspect, adisclosed transmitter for wireless communication includes a firstplurality of antennas, a symbol mapper configured to map an input blockof an incoming binary bit stream representing information to betransmitted to a symbol representing an ordered plurality of complexnumbers, the input block comprising a plurality of binary bits, aspace-time encoder configured to apply an encoding operator to thesymbol to produce a vectorized space-time codeword defining electricalsignals to be transmitted by the transmitter, the encoding operatorbeing dependent on a plurality of predefined stabilizer generators, andcircuitry to collectively transmit, by the first plurality of antennasto a second plurality of antennas of a receiver over a wirelesstransmission channel, the electrical signals defined by the vectorizedspace-time codeword.

In another aspect, a disclosed method for transmitting information in awireless communication system includes receiving an incoming binary bitstream representing the information, and for each input block of theincoming binary bit stream comprising a predefined number of binarybits, mapping the input block to a respective symbol representing anordered plurality of complex numbers, applying an encoding operator tothe respective symbol to produce a respective vectorized space-timecodeword defining electrical signals to be transmitted, the encodingoperator being dependent on a plurality of predefined stabilizergenerators, and transmitting, collectively and by a first plurality ofantennas of a transmitter to a second plurality of antennas of areceiver over a wireless transmission channel, the electrical signalsdefined by the vectorized space-time codeword.

In yet another aspect, a disclosed system for multiple-inputmultiple-output (MIMO) transmission includes a transmitter including afirst plurality of antennas and a receiver including a second pluralityof antennas. The transmitter further includes a symbol mapper configuredto map an input block of an incoming binary bit stream representinginformation to be transmitted to a symbol representing an ordered pairof complex numbers, the input block comprising a plurality of binarybits, a space-time encoder configured to apply an encoding operator tothe symbol to produce a vectorized space-time codeword definingelectrical signals to be transmitted by the transmitter, the encodingoperator being dependent on a plurality of predefined stabilizergenerators, and circuitry to collectively transmit, by the firstplurality of antennas of the transmitter to the second plurality ofantennas of the receiver over a wireless transmission channel, theelectrical signals defined by the vectorized space-time codeword.

In any of the disclosed embodiments, the receiver may include aspace-time decoder configured to recover the symbol from the electricalsignals transmitted by the transmitter using a decoding operation thatis based on maximum likelihood inference, and a symbol de-mapperconfigured to recover the input block from the symbol. The receiver mayalso include circuitry to collectively receive, by the second pluralityof antennas of the receiver from the first plurality of antennas of thetransmitter over the wireless transmission channel, the electricalsignals defined by the vectorized space-time codewords.

In any of the disclosed embodiments, the wireless transmission channelmay be a noncoherent multiple-input multiple-output (MIMO) channelcharacterized by an unknown channel matrix and may be assumed to havecoherence over a predefined time period and additive Gaussian noise.

In any of the disclosed embodiments, the number of antennas in the firstplurality of antennas and the number of antennas in the second pluralityof antennas may be the same.

In any of the disclosed embodiments, the number of antennas in each ofthe first plurality of antennas and the second plurality of antennas maybe equal to a power of two.

In any of the disclosed embodiments, the number of antennas in each ofthe first plurality of antennas and the second plurality of antennas maybe two.

In any of the disclosed embodiments, the symbol may represent an orderedpair of complex numbers.

In any of the disclosed embodiments, each stabilizer generator S havingan index value l of the plurality of predefined stabilizer generators isdefined as follows:

$S_{l} = {X \otimes I_{2^{\lfloor\frac{}{2}\rfloor}} \otimes X \otimes I_{2^{k - 1}} \otimes X \otimes I_{2^{k - 1 - {\lfloor\frac{}{2}\rfloor}}}}$

when l is even,

$S_{l} = {X \otimes I_{2^{\lfloor\frac{}{2}\rfloor}} \otimes Z \otimes I_{2^{k - 1}} \otimes Z \otimes I_{2^{k - 1 - {\lfloor\frac{}{2}\rfloor}}}}$

when l is odd, l may range from 0 to one less than twice the base-twologarithm of the number of antennas in the first plurality of antennas,I may represent an identity matrix, and X and Z may represent respectivePauli matrices defined as follows:

${X = \begin{pmatrix}0 & 1 \\1 & 0\end{pmatrix}},{Z = {\begin{pmatrix}1 & 0 \\0 & {- 1}\end{pmatrix}.}}$

In any of the disclosed embodiments, the symbol mapper may be dependenton a target data rate for transmission of the information.

BRIEF DESCRIPTION OF THE DRAWINGS

Embodiments of the present disclosure may be better understood throughreference to the following figures in which:

FIG. 1 is a block diagram illustrating selected elements of a wirelesscommunication system 100, according to one embodiment;

FIG. 2 is a block diagram illustrating a MIMO wireless communicationsystem, in accordance with some embodiments;

FIG. 3 illustrates selected elements of an example method for space-timeencoding in a wireless communication system in flow diagram form,according to one embodiment;

FIG. 4 illustrates selected elements of an example method for encodingsymbols using space-time encoding in flow diagram form, according to oneembodiment;

FIG. 5 illustrates selected elements of an example method for space-timedecoding in flow diagram form, according to one embodiment;

FIGS. 6A and 6B illustrate an example application of a projection matrixand error operators, in accordance with some embodiments;

FIG. 7 illustrates a graph plotting bit error probability vs. SNR forthe disclosed space-time encoding techniques as well as other encodingtechniques;

FIGS. 8A and 8B illustrate results of simulations in which the errorprobability is bounded, according to some embodiments; and FIG. 9illustrates an example computing system for space-time encoding in awireless communication system, according to one embodiment.

DETAILED DESCRIPTION OF PARTICULAR EMBODIMENT(S)

In the following description, details are set forth by way of example tofacilitate discussion of the disclosed subject matter. It should beapparent to a person of ordinary skill in the field, however, that thedisclosed embodiments are exemplary and not exhaustive of all possibleembodiments. For a more complete understanding of the presentdisclosure, reference is made to the following description andaccompanying drawings.

Disclosed herein are methods for encoding and decoding information fortransmission over wireless channels with multiple transmit and receiveantennas. In at least some embodiments, the disclosed methods ofdecoding do not rely on an estimate of the wireless channel. Theconstruction is amenable to a variety of code rates and has minimallatency for a given asymptotic high-SNR performance.

In at least some embodiments, the space-time block code described hereinfor noncoherent communication may apply techniques from the field ofquantum error correction. For example, a multiple-input multiple-output(MIMO) channel may be decomposed into operators used in quantummechanics, and a noncoherent space-time code may be designed using thequantum stabilizer formalism. An optimal space-time decoder may bederived in accordance with the space-time encoders described herein.This approach has been compared to a coherent space-time coding approachand a noncoherent differential approach, achieving comparable or betterperformance.

Space-time codes are a family of methods employed to increase thechannel diversity of a communication system by re-transmittinginformation over multiple spatial and time degrees of freedom.Communication approaches leveraging diversity may provide increasedreliability by sending information over multiple independent channelrealizations. This results in a higher chance of being able to correctlydecode the received signal. In at least some embodiments, the techniquesdescribed herein may protect against deep fade events, also known asoutages, which contribute the largest source of symbol decoding errorsin very slowly varying, rich scattering environments like Rayleighfading.

As noted above, the wireless communication systems described herein mayemploy a specific noncoherent space-time block code (and a correspondingdecoder) that was designed using techniques inspired by quantum errorcorrecting codes. A noncoherent space-time code may be defined as aspace-time code that is designed for use in a wireless communicationsystem that does not explicitly estimate the channel. However, such acode may also be used in a wireless communication system that doesestimate the channel. These techniques may be used to design codes ofvarious data rates (i.e., the number of bits transmitted per second perHz of bandwidth). The space-time block code described herein wasdesigned, in part, by drawing a comparison between noncoherent wirelesschannels and quantum channels and applying a code for an idealizedquantum channel to a non-ideal wireless channel. Through numericalsimulations, it has been found that these codes result in improvedperformance (in terms of bit error rate) in the low-data-rate regimewith respect to other noncoherent and coherent space-time block-codeconstructions with analogously short coherence times. In someembodiments, the disclosed techniques may be used to reliablycommunicate information over multiantenna channels with minimal latencyat low rates, which may be especially useful in the regime ofultra-reliable low-latency communication or in tactical communicationenvironments.

FIG. 1 is a block diagram illustrating selected elements of a wirelesscommunication system 100, according to one embodiment. In this example,a transmitter 115 including multiple transmitting antennas 112 isinstalled on a drone 110, and a receiver 135 including multiplereceiving antennas 132 is included in a mobile communication device 130.The wireless communication channel 120 illustrated in FIG. 1 may be aMIMO channel characterized by an unknown channel matrix. In thisexample, information to be transmitted between transmitter 115 andreceiver 135 over wireless communication channel 120 may be transmittedcollectively by the multiple transmitting antennas 112 and receivedcollectively by the multiple receiving antennas 132 using the space-timeencoding techniques described herein to protect against certain types oferrors. For example, these techniques may protect data fromenvironmental distortion (i.e., noise) and/or deep fade events, asdescribed herein. In some embodiments, these techniques may be appliedin wireless communication systems that experience frequency hopping orother scenarios in which the coherence time is short, such as highmobility environments.

Unlike existing coding techniques that utilize estimates of thecommunication environment to transmit and receive information, quantumchannel coding and noncoherent space-time coding do not rely onestimating the channel. Instead, these code designs leverage fewassumptions about the channel. For example, in some embodiments, thechannel may be assumed to have coherence over a predefined time periodand additive noise, such as Gaussian noise. In at least someembodiments, the coherence interval, in terms of channel uses, may beapproximately 2 times the number of antennas in the transmitter.

A block-fading channel model assumes that the channel coefficientsremain constant for a relatively short block of consecutive symbols butmay change for the next block. In other words, the block fading modelassumes extremely short coherence times.

As described in more detail herein, a code for noncoherent communicationhas been designed based on quantum stabilizer codes. The design of thecode may begin with a decomposition of the communication channel intoelements of the Pauli group, a well-known matrix basis from quantummechanics. The transmitted information is viewed as a quantum state andthe code is designed to mitigate the effects of the channel under theassumption of infinite signal-to-noise ratio (SNR). An optimal decodingrule is then derived for the noisy case.

The theory of quantum error correction (QEC) defines conditions allowingfor the correction of a broad class of channel errors. Many families ofQEC codes were developed by directly extending classical errorcorrecting codes. For example, Calderbank-Shor-Steane (CSS) codesgeneralize self-dual classical codes to the quantum setting. A CSSconstruction based on low-density parity-check (LDPC) codes, which arelinear error correcting codes, was developed for protecting large blocksof qubits against noise. The space-time coding techniques describedherein approach the problem from the opposite direction, using quantummechanical concepts to inspire algorithms for classical communication inthe noncoherent setting.

Estimation of the capacity of the noncoherent channel has been derivedfor particular cases, as described in the literature. The capacityachieving code construction performs sphere packing on the Grassmannmanifold, leading to codes using Grassmannian line packings. Adifferential encoding based on matrix groups has been proposed thatimplicitly performs channel estimation, although the estimate is updatedusing only data from two coherence intervals. Differential coding hasalso been extended to matrix families that are not groups.

In one earlier approach, a subspace code, namely a Grassmannian linepacking for noncoherent communication, was derived using formalism fromthe Pauli group and stabilizer coding. While the present approachsimilarly exploits both Grassmannian line packings and quantumstabilizer coding, the approach to the coding problem described hereindiffers from the earlier approach in several significant ways. In oneexample, the previous approach included a construction for Grassmannianline packings designed to maximize spectral efficiency (i.e., the rateof communication per unit of bandwidth). In practice, such constructionsmay only be useful in the long blocklength regime. In another example,the previous approach is a construction based on operator Reed-Muellercodes over the Pauli Group. It is not optimized for optimal performanceat high signal to noise ratio, nor in terms of the symbol error rateperformance metric. In contrast, the approach presented herein proposesto (directly) use codewords from a particular set quantum stabilizercodes as codewords in a space-time block code. The resulting code isparticularly well suited for use at short blocklengths and is optimizedfor high SNR performance.

Notation: In the descriptions that follow, bold lower-case letters, suchas a, are typically used to denote column vectors, and bold upper-caseletters, such as A, are typically used to denote matrices. Non-boldletters are typically used to denote scalars. The element in the i^(th)row and k^(th) column of a matrix A is denoted by [A]_(i,k). In general,the k×k identity matrix is denoted by I_(k). The 2×2 identity matrix isused so often that the subscript is often dropped, i.e. I₂=I. Thenotation tr(A) is used to denote the trace, det(A) the determinant,A^(T) the transpose, and A* the conjugate transpose. For positivesemi-definite matrices,

$A^{\frac{1}{2}}$

denotes the matrix square root. The notation |a| is used to denote theabsolute value of a scalar. The notation

(·) is used to denote expectation. The symbol ⊗ is used to denote thetensor product when acting on vector spaces (i.e.

²⊗

²) and to denote the Kronecker product when acting on vectors ormatrices. Note that the Kronecker product is an operation on twomatrices of arbitrary size resulting in a block matrix. The notation

(μ,Σ) is used to denote a complex circularly symmetric normaldistribution with mean μ and covariance Σ. If A=cB where c>0, this isdenoted as A ∝B.

Qubits, or quantum bits, are the natural generalization of a bit inquantum computing. A qubit represents the state of a two-level system,such as the polarization of a photon, and is the most elementary exampleof quantum state. A qubit is represented as a state vector q=[α,β]^(T)∈

² with q*q=|α|²+|β|²=1 by convention. Equipping

² with the standard inner product

q, p

=q*p leads to a definition of a qubit state as an element of atwo-dimensional complex Hilbert space. In discrete time, the evolutionof a closed quantum system is unitary. That is to say, q_(n+1)=Uq_(n),where U∈

^(2×2) with U*U=I. The concept of applying unitary operators (which ispossible to good approximation) comes up often in quantum computing andmay be used in the space-time coding described herein.

In general, the act of measuring, or observing, a quantum state causesthe system to change. An important class of quantum measurements areprojective measurements. Projective measurements are defined in terms ofan observable, which is a Hermitian operator M on the state space of thesystem of interest. Let m denote an eigenvalue of M and let P_(m) be theprojector onto the m eigenspace. The observable M can thus beorthogonally diagonalized as

$M = {\sum\limits_{m}{mP_{m}}}$

where P_(n)P_(m)=0 when n≠m. The outcome of “measuring the observable M”is an eigenvalue m. If ψ is a quantum state, then the probability ofmeasuring m is given by p(m)=ψ*P_(m)ψ. Given that the outcome m occurs,the system after measurement collapses to the state P_(m)ψ/√{square rootover (p(m))}. It has been discovered that projective measurements,coupled with unitary evolution, may be used to fully describe generalquantum measurements.

A notable feature of quantum measurement is that the global phase of astate is not observable. If x=[α,β]^(T) and y=e^(jθ)[α,β]^(T), then, fora measurement in all possible bases, the distributions of outcomes for xand y are the same. For this reason, one often works with densitymatrices. A state q can be represented by its density matrix Q=qq*. Thepreviously described formalism can be represented analogously. A state Qthat evolves by the unitary U becomes the state UQU*. The Born rule forprojective measurements says that a state Q evolves to P_(m)QP_(m)/p(m)with probability p(m)=Tr(P_(m)QP_(m)). Density matrices may additionallyprovide a convenient way to describe quantum systems that have classicaluncertainty. If a system is prepared in the state ψ with probabilityp_(i), then the system may be represented by the density matrix, asshown below:

Q=Σ _(i) p _(i)ψ_(i)ψ_(i)*

This example can be extended to the case in which the prepared state hasa continuous distribution and ensures that measurement probabilities areproperly modeled. A state with a rank-one density matrix is known as apure state and corresponds to the case in which there is no classicaluncertainty about the prepared state. A state with a higher-rank densitymatrix is known as a mixed state.

Systems of many qubits can be represented as extensions of a singlequbit system with the tensor and Kronecker products. The state space ofan n-qubit system is the tensor product of the n component single qubitsystems, i.e.

² ^(n) =

²⊗

² . . . ⊗

². Any normalized vector in

² ^(n) is a valid state vector. For example, if q₁ and q₂ are singlequbit systems, the two-qubit composite system may be given by q₁⊗q₂.Analogously, if Q₁ and Q₂ are density matrix representations of twosystems, the composite system may have a density matrix of Q₁⊗Q₂. Anypositive semidefinite operator with a trace equal to unity may be avalid density operator. In multi-qubit systems, observables andmeasurements may be defined in the relevant higher dimensional space.

The fidelity provides a notion of distance between quantum states. Thefidelity function F(Q₁, Q₂)∈[0,1] is a symmetric function of its densitymatrix arguments. It is defined, for general mixed states as

${F\left( {Q_{1},Q_{2}} \right)} = {{{tr}\left( \left( {Q_{1}^{\frac{1}{2}}Q_{2}Q_{1}^{\frac{1}{2}}} \right)^{\frac{1}{2}} \right)}.}$

A low fidelity implies that states are “far apart,” and the fidelity isequal to unity if its arguments are the same. The fidelity between twopure states is F(q₁, q₂)=|q*₁q₂|. The fidelity between a pure state qand a mixed state Q is F(q,Q)=√{square root over (q*Qq)}. The fidelitycan be used to induce a metric on states, d, via d(Q₁, Q₂)=arccos(F(Q₁,Q₂)).

Stabilizer codes are a class of quantum error correcting codes designedto protect against a wide range of quantum errors. Their construction isbriefly summarized below.

The n-qubit Pauli group, P_(n), is the set of operators in

² ^(n) ^(×2) ^(n) that can be written as a tensor product of n of the2×2 Pauli matrices I, X, Y, Z, up to a scalar multiple of α∈{±1, ±j},where

${X = \begin{pmatrix}0 & 1 \\1 & 0\end{pmatrix}},{Y = \begin{pmatrix}0 & {- j} \\j & 0\end{pmatrix}},{Z = {\begin{pmatrix}1 & 0 \\0 & {- 1}\end{pmatrix}.}}$

The group multiplication operation is defined as standard matrixmultiplication. Elements of P_(n) are unitary and are either Hermitianor skew-Hermitian. Thus, they are orthogonally diagonalizable, witheigenvalues±1 or ±j. Any two Pauli operators either commute oranti-commute.

A stabilizer, S, is a commutative (Abelian) subgroup of P_(n) that doesnot contain the negative of the identity element, −I₂ _(n) . Theadditional requirement of closure under the group multiplicationoperation implies that elements of S must have α=1. Thus, elements of Sare Hermitian operators with eigenvalues equal to ±1.

A stabilizer code C(S) is defined as the subspace of

² ^(n) formed by the intersection of +1 eigenspaces of the operators inS. An efficient description of the group S is its generators. A setG_(S) of generators of S is a set of elements of P_(n) such that everyelement of S is a product of elements in G_(S). A generator set G_(S) iscalled independent if the set obtained by removing an element from G_(S)fails to generate all elements of S. If S is a stabilizer with anindependent generator containing n−k elements, it can be shown that C(S)is a 2^(k) dimensional vector space. Furthermore, a state ψ∈C(S) if andonly if S_(n)ψ=ψ for all S_(n)∈G_(S). Letting s_(i) denote complexconstants and v_(i) an orthonormal basis for C(S), a general codewordfor C(S) can be written as

${x = {\sum\limits_{i = 0}^{2^{k} - 1}{s_{i}v_{i}}}},{{{with}\mspace{20mu} {\sum\limits_{i = 0}^{2^{k} - 1}{s_{i}}^{2}}} = {1.}}$

A codeword is thus an arbitrary unit vector in C(S).

There are several criteria that can be used to determine which quantumerrors a stabilizer code can correct. One approach is as follows.Consider a set of error operators E⊂P_(n). Each error E∈E eithercommutes or anti-commutes with each generator of the stabilizer group. Asufficient condition for the stabilizer code to correct the errors in Eis for each E∈E to possess a unique commutation relationship withrespect to the elements of G_(S). Thus, the stabilizer constructionguarantees that each error E_(i)∈E maps the codespace C(G) bijectivelyto a 2^(k) dimensional subspace of

² ^(n) . Furthermore, the uniqueness of the commutation relationshipsguarantees that different errors map C(G) to different error subspacesε_(i). Formally, ε_(i) is the image of E restricted to C(G) (i. e. ε_(i){y∈

² ^(n) |∃x∈C(G) with y=E_(i)x}) and a unique commutation relationshipguarantees that ε_(i)∩ε_(j)=Ø when i≠j.

It should be noted that this criterion is sufficient but not necessary;the stabilizer formalism naturally lends itself to degenerate quantumcodes, where multiple errors yield the same syndrome and are correctableby the same operation. Consider a correctable error E and V∈

. Both EV and E will have the same commutation relations with respect tothe stabilizer generators, and thus both EV and E map an encoded stateto the same subspace. Indeed, for x∈C(G), EVx=Ex; namely, the effect ofthe errors on the codeword is exactly the same.

In the quantum setting, the stabilizer decoding process consists ofperforming projective measurements on the received state. Themeasurement observables are the stabilizer generators. This processprojects the state into an intersection of the +1 or −1 eigenspaces ofeach S∈G(S). Thus, after the measurements the state collapses into oneof the error subspaces ε_(k). The measurement outcomes form a syndrome(analogous to the classical syndrome) and identify into which subspacethe state collapsed. The error correction conditions guarantee that theapplication of a correction (e.g., the error operator itself) for anycorrectable error yielding the measured syndrome recovers the encodedstate. This process demonstrates that a stabilizer code that can correcterrors in a set E can correct an arbitrary linear combination ofcorrectable errors.

In some embodiments, projection operations may annihilate erroroperators that are not consistent with the measured commutationrelationship. For example, consider the state y=(c_(j)E_(j)+c_(k)E_(k))twith t∈C(S). If G∈G_(S) anti-commutes with an error E_(j) but commuteswith E_(k) and a measurement of G returns a 1 (corresponding to acommutation), the state after measurement is ŷ=(I+G)y=E_(k)t.

In the descriptions that follow, a specific canonical received signalmodel is assumed for noncoherent wireless communication. The systemincludes N_(TX)=M1, N_(RX)=M2 antennas, respectively, at the transmitterand receiver. In at least some embodiments M1=M2. A narrowband modelwith a single-tap (frequency flat) MIMO channel represented as H∈

^(M1×M2) is assumed. It is also assumed that the channel is coherent forTC channel uses. The transmitted, received, and additive noise signalsare denoted by the complex matrices T∈

^(M1×TC), Y∈

^(M2×TC) and N∈

^(M2×TC), respectively, where the columns correspond to the timeinstants in the coherence interval. In this system model, N may beassumed to be a complex Gaussian random matrix with independent,identically distributed entries such that [N]_(i,j)˜Nc(0,σ_(n) ²).Furthermore, a Rayleigh fading model may be assumed, where the entriesof H are independent and identically distributed with [H]_(i,j)·Nc(0,1).The vectorized fading channel may be represented as:

$\overset{\_}{H} = {{\sum\limits_{j = 0}^{N_{ant}^{2}}{c_{j}E_{j}}} = {I_{T} \otimes {\sum\limits_{F_{j} \in P_{{k_{1}{P_{k}}} = {4^{k} = N_{ant}^{2}}}}{c_{j}F_{j}}}}}$

Finally, it is assumed that H is constant over the coherence intervalbut that the channel realizations at different coherence intervals areindependent. This model may be especially appropriate for a frequencyhopping system in an environment with rich scattering. The receivedsignal over the coherence interval is now given by:

Y=HT+N.

Using the standard vectorization identity, letting y=vec(Y)∈

^(T2) ^(k) , t=vec(T)∈C^(T2) ^(k) , n=vec(N)∈

^(T2) ^(k) , and H=I_(T)⊗H, the equation above may be written as:

y=Ht+n.

It has been discovered that if M=2^(k) for some integer k andT=2^(r)≥2M, this form of the channel model is amenable to the design ofa stabilizer code.

FIG. 2 is a block diagram illustrating a MIMO wireless communicationsystem 200 including a transmitter 210, a channel 230, and a receiver240. In the illustrated example, transmitter 210 includes, among otherelements, a symbol mapper 215 configured to map an input block 205 of anincoming binary bit stream representing information to be transmitted toa symbol representing an ordered plurality of complex numbers, the inputblock including a predetermined number of binary bits, and a space-timeencoder 220 configured to apply an encoding operator to the symbol toproduce a vectorized space-time codeword defining electrical signals tobe transmitted by the transmitter, the encoding operator being dependenton a plurality of predefined stabilizer generators. Symbol mapper 215may map each input block 205 containing a predetermined number of bitsto a symbol, which may be viewed as analogous to a qubit, represented byan ordered pair of complex vectors of modulus one. The symbol may bechosen from a set of symbols in a constellation, where there is a 1-to-1mapping between possible input blocks and available symbols. Theconstellation may be dependent on a modulation format employed in thetransmission of information in wireless communication system 200. Insome embodiments, the magnitude and phase of each complex number mayrepresent the in-phase [I] and quadrature [Q] components of a basebandsignal. Space-time encoder 220 may then map the symbol, which isessentially a complex 2-vector, to a symbol in a higher dimensionalspace, such as to an 8-vector space-time symbol, in some embodiments, byencoding the symbol using a space-time codeword.

In at least some embodiments, the number of antennas in the transmitterand the number of antennas in the receiver may be the same. In at leastsome embodiments, the number of antennas in each of the transmitter andthe receiver may be equal to a power of two. However, the techniquesdescribed herein may be applicable in wireless communication systems inwhich the number of antennas in the transmitter and the number ofantennas in the receiver are not the same and/or the number of antennasin the transmitter or in the receiver is not equal to a power of two. Insome embodiments, MIMO wireless communication system 200 may be a 2×2MIMO system in which transmitter 210 includes two transmitting antennasand receiver 240 includes two receiving antennas.

Transmitter 210 may also include circuitry to collectively transmit, bya first plurality of antennas of the transmitter to a second pluralityof antennas of a receiver over a wireless transmission channel, theelectrical signals defined by the vectorized space-time codeword (notshown in FIG. 2). For example, the 8-vector space-time symbol mayrepresent the electrical signals to be transmitted over four timeinstances, or four uses, of a two-by-two MIMO channel. In theillustrated example, channel 230 may be a MIMO channel characterized byan unknown channel matrix H, as described herein.

In the illustrated example, receiver 240 includes, among other elements,a space-time decoder 245 configured to recover the symbol from theelectrical signals transmitted by the transmitter using a decodingoperation that is based on maximum likelihood inference, and a symbolde-mapper 250 configured to recover the input block 205 from the symbolas recovered input block 255. In other words, space-time decoder 245 mayreturn an estimate of the symbol transmitted by transmitter 210. Symbolde-mapper 250 may then demodulate the estimated symbol to recover thebits of the input block.

According to the assumptions described herein for a wirelesscommunication system, a model for transmitting a single coherent blockin the system may be shown as:

$\begin{pmatrix}Y_{1,1} & \ldots & Y_{1,{TC}} \\\vdots & \ddots & \vdots \\Y_{{M2},1} & \ldots & Y_{{M\; 2},{TC}}\end{pmatrix} = {{\begin{pmatrix}H_{1,1} & \ldots & H_{1,{M\; 1}} \\\vdots & \ddots & \vdots \\H_{{M2},{TC}} & \ldots & H_{{M\; 2},{TC}}\end{pmatrix}\begin{pmatrix}X_{1,1} & \ldots & X_{1,{TC}} \\\vdots & \ddots & \vdots \\X_{{M\; 1},1} & \ldots & X_{{M\; 1},{TC}}\end{pmatrix}}\  + \begin{pmatrix}N_{1,1} & \ldots & N_{1,{TC}} \\\vdots & \ddots & \vdots \\N_{{M2},1} & \ldots & N_{{M\; 2},{TC}}\end{pmatrix}}$

A vectorized form of this model may be shown as.

$\begin{pmatrix}\begin{bmatrix}Y_{1,1} \\\vdots \\Y_{{M\; 2},1}\end{bmatrix} \\\vdots \\\begin{bmatrix}Y_{1,{TC}} \\\vdots \\Y_{{M\; 2},{TC}}\end{bmatrix}\end{pmatrix} = {{\left( {\begin{matrix}\begin{pmatrix}H_{1,1} & \ldots & H_{1,{M\; 1}} \\\vdots & \ddots & \vdots \\H_{{M2},1} & \ldots & H_{{M\; 2},{M\; 1}}\end{pmatrix} & \ddots \\0 & \;\end{matrix}\begin{matrix}0 \\\begin{pmatrix}H_{1,1} & \ldots & H_{1,{M\; 1}} \\\vdots & \ddots & \vdots \\H_{N,1} & \ldots & H_{{M\; 2},{M\; 1}}\end{pmatrix}\end{matrix}} \right)\begin{pmatrix}\begin{bmatrix}X_{1,1} \\\vdots \\X_{{M\; 1},1}\end{bmatrix} \\\vdots \\\begin{bmatrix}X_{1,{TC}} \\\vdots \\X_{{M\; 1},{TC}}\end{bmatrix}\end{pmatrix}} + \begin{pmatrix}\begin{bmatrix}N_{1,1} \\\vdots \\N_{{M\; 2},1}\end{bmatrix} \\\vdots \\\begin{bmatrix}N_{1,{TC}} \\\vdots \\N_{{M\; 2},{TC}}\end{bmatrix}\end{pmatrix}}$

In at least some embodiments, the wireless communication system mayinclude two antennas at both the transmitter and the receiver. In onesuch embodiment, a narrowband model with a single-tap MIMO channel H∈

^(2×2) is assumed, as is a channel coherence time of TC=4 channel uses.The transmitted, received, and additive noise signals may be denoted bycomplex 2×4 matrices T, Y, and N, respectively, of the form definedabove, where the columns correspond to the time instants in thecoherence interval.

A vectorized model for this specific system may be shown as:

$\begin{pmatrix}\begin{bmatrix}Y_{11} \\Y_{21}\end{bmatrix} \\\begin{bmatrix}Y_{12} \\Y_{22}\end{bmatrix} \\\begin{bmatrix}Y_{13} \\Y_{23}\end{bmatrix} \\\begin{bmatrix}Y_{14} \\Y_{24}\end{bmatrix}\end{pmatrix} = {{\begin{pmatrix}\begin{pmatrix}H_{11} & H_{12} \\H_{21} & H_{22}\end{pmatrix} & \; & 0 & \; \\\; & \begin{pmatrix}H_{11} & H_{12} \\H_{21} & H_{22}\end{pmatrix} & \; & \; \\\; & \; & \begin{pmatrix}H_{11} & H_{12} \\H_{21} & H_{22}\end{pmatrix} & \; \\\; & 0 & \; & \begin{pmatrix}H_{11} & H_{12} \\H_{21} & H_{22}\end{pmatrix}\end{pmatrix}\begin{pmatrix}\begin{bmatrix}X_{11} \\X_{21}\end{bmatrix} \\\begin{bmatrix}X_{12} \\X_{22}\end{bmatrix} \\\begin{bmatrix}X_{13} \\X_{23}\end{bmatrix} \\\begin{bmatrix}X_{14} \\X_{24}\end{bmatrix}\end{pmatrix}} + \begin{pmatrix}\begin{bmatrix}N_{11} \\N_{21}\end{bmatrix} \\\begin{bmatrix}N_{12} \\N_{22}\end{bmatrix} \\\begin{bmatrix}N_{13} \\N_{23}\end{bmatrix} \\\begin{bmatrix}N_{14} \\N_{24}\end{bmatrix}\end{pmatrix}}$

An example method 300 for space-time encoding in a wirelesscommunication system is illustrated by the flow diagram in FIG. 3,according to one embodiment. As illustrated at 302, the method mayinclude receiving, at a transmitter of a wireless communication system,an incoming binary bit stream representing information to betransmitted.

At 304, the method may include mapping a given input block of theincoming binary bit stream, having a predefined number of bits, to asymbol representing a plurality of complex numbers. The symbol may bechosen from a set of 2^(q) symbols in a constellation, where q is thenumber of bits in each input block, such that there is a 1-to-1 mappingbetween possible input blocks and available symbols. In at least someembodiments, each symbol s may include an ordered pair of complexnumbers is of the form

$s = {\begin{bmatrix}\alpha \\\beta\end{bmatrix}.}$

As described in more detail herein, the constellation of symbols may bebased on a Grassmannian line packing. In at least some embodiments, thesymbol mapper may be dependent on a target data rate for transmission ofthe information. The target data rate r may be equal to the number ofbits in each input block divided by the coherence interval. Theavailable code rates, R (in bits/second/Hz) may be defined by:

${R = \frac{\log_{2}\left( {C} \right)}{2N_{ant}}}.$

At 306, method 300 may include applying an encoding operator to thesymbol to produce a vectorized space-time codeword defining electricalsignals to be transmitted, the encoding operator being dependent on aplurality of predefined stabilizer generators. As described in moredetail herein, the encoding causes a linear mapping t=Cs from the symbols to a respective codeword t in a codeword space. In at least someembodiments, the codeword space is a 2^(m) ² ×2 matrix defined via thepredefined stabilizer generators, where m is the number of antennas inthe transmitter. The codeword t is a 2^(m) ² vector.

At 308, the method may include transmitting, collectively and by a firstplurality of antennas of the transmitter to a second plurality ofantennas of a receiver over a wireless transmission channel, theelectrical signals defined by the vectorized space-time codeword. Morespecifically, the first m elements of t are a complex basebandrepresentation of the electrical signals to be transmitted using the mtransmitting antennas at the first time instant, the second m elementsof t are the complex baseband representation of the electrical signalsto be transmitted at the second time instant, and so on. At each timeinstant, the m elements of t may be upconverted before beingtransmitted.

Designing a quantum stabilizer code for the space-time channel requiresa representation of the channel model analogous to a quantum operation.For example, t may be modeled as an encoded quantum state. Like theirclassical linear counterparts, a quantum code is a linear subspace.Quantum states are encoded in higher dimensional quantum systems via alinear map. The generator matrix for a code that maps

qubits to m is defined as the (tall) matrix C∈

whose columns form an orthonormal basis for the code. The state s∈

is encoded into t∈C² ^(m) via

t=Cs.

Substituting this into the earlier equation yields:

v=HCs+n.

A “pure” quantum state is a normalized vector defined up to a “globalphase”. Notably, two pure states {circumflex over (ψ)} and ψ areequivalent by definition if {circumflex over (ψ)}{circumflex over(ψ)}*={circumflex over (ψ)}{circumflex over (ψ)}*. Thus, a pure staterepresents a one-dimensional subspace of the (

dimensional) state space. Formally, a state is a normalizedrepresentation of a point on the Grassmann manifold GF (1,

). In the descriptions herein, s is interpreted as a quantum state and agoal is to design a code that recovers an estimate s such that, atinfinite SNR, ŝŝ*=ss*. The vector s may represent the transmittedsymbol. The first step in the modulation may be to map bits to someconstellation C of vector symbols s with constant modulus, which may beviewed as representing a quantum state to be encoded. Note that sincetwo quantum systems y and x are equivalent if xx*=yy* the constellationmay be thought of as an alphabet of quantum states. The constellation Cmay be chosen as a Grassmannian line packing in

dimensions, which is later shown to minimize an upper bound on theprobability of error. While noncoherent space-time codes may be designedvia Grassmannian line packings of the row space of T, the packing usedto design C may be in a lower dimensional space.

For models with zero-mean noise that is independent of the channel, areasonable choice for an “emulated” quantum channel, ε, may be asfollow:

${tt^{*}}\overset{\; ɛ\mspace{11mu}}{\rightarrow}{{\left( {vv}^{*} \middle| t \right)}.}$

This map is both completely positive (by definition) and is convexlinear in tt*. Thus, it may be used to form a valid quantum channel upto a normalizing constant. Via Choi's Theorem, for some set of so-called“error operators” E, this may be written as:

ε(tt*)=Σ_(E) _(i) _(∈E) E _(i) tt*E _(i)*.

The construction of an emulated quantum channel model for theN_(TX)=N_(RX)=2 case is described in more detail below.

Given a quantum code C designed to protect against some set of errors,one can construct a “recovery operation” that is a physically realizablequantum operation mapping a noisy state back into the codespace. If thecode is such that all possible errors in some channel are correctable,then the recovery operation is guaranteed to map the noisy channeloutput back to its input. This recovery operation can be written as aquantum operation via

R _(C)(p)=Σ_(i) E _(i) * P _(i) pP _(i) E _(i)

for a set of projectors P_(i) and a set of correctable errors E_(i)which may, in general, be linear combinations of the elements of E_(i).

A code's emulated recovery operation may be defined as a mapping frompositive matrices to positive matrices given by:

f _(R) _(C) (P)=Σ_(i) E _(i) ^(H) P _(i) pP _(i) E _(i)

It has been discovered that if a channel induces an emulated channeldescribed by a set of error operators E that are correctable by somecode C, this may yield, for a random real positive constant c:

${C^{*}{f_{R_{C}}\left( {vv}^{*} \right)}C}\overset{a.s.}{=}{{c\left( {ss}^{*} \right)}.}$

Since the input constellation C is constant modulus, the Cauchy-Schwarzinequality, for an input symbol s∈C, yields:

$s\overset{a.s.}{=}{\begin{matrix}{\arg \mspace{14mu} \max} \\{\hat{s} \in C}\end{matrix}{\hat{s}}^{*}C^{*}{f_{_{C}}\left( {vv}^{*} \right)}C{\hat{s}.}}$

It has been discovered that, in the finite SNR case, it is impossible todesign a quantum code to correct all of the possible errors that arise.Therefore, a code is constructed assuming infinite SNR and the optimaldecoder for the finite SNR case is derived therefrom. It has beendiscovered that, in a setting that includes a linear Gaussian channelwith additive Gaussian noise, the maximum likelihood (ML) decoding rulefor finite SNR is given by the equation shown in the immediatelypreceding paragraph.

An example construction of a noncoherent space-time code via thestabilizer formalism in a wireless communication system including 2antennas at both the transmitter and the receiver is described in moredetail below. The application of quantum error correcting codes in awireless communication setting may be motivated by noting that at highSNR, the vectorized channel model may be well approximated by astochastic linear combination of Pauli group elements, i.e., Paulimatrices.

The vectorized channel matrix H highlights the coherence of the channelcoefficients over time and admits a basis decomposition in the Paulibasis P₃ of the form

$\begin{matrix}{\overset{\_}{H} = {I \otimes I \otimes \left( {{c_{0}I} + {c_{1}X} + {c_{2}Z} + {c_{3}Y}} \right)}} \\{= {{c_{0}E_{0}} + {c_{1}E_{1}} + {c_{2}E_{2}} + {c_{3}E_{3}}}}\end{matrix}$ where c₀ = ([H]_(1, 1) + [H]_(2, 2))/2c₁ = ([H]_(1, 2) + [H]_(2, 1))/2 c₂ = ([H]_(1, 1) − [H]_(2, 2))/2c₃ = j([H]_(1, 2) − [H]_(2, 1))/2

and E₀=I⊗I⊗I, E₁=I⊗I⊗X, E₂=I⊗I⊗Z, E₃=I⊗I⊗Y.

Defining c=[c₀, c₁, c₂, c₃]^(T) yields c˜

(0, I₄/2). Note that at infinite SNR, the vectorized channel model maybe written as

v=Ht=HCs=(c ₀ I _(T) ⊗I+c ₁ I _(T) ⊗X+c ₂ I _(T) ⊗Z+c ₃ ⊗Y)t.

The error set for this channel is E={E₀, E₁, E₂, E₃}. This process isanalogous to the quantum concept of channel discretization, in which achannel with a continuous set of possible realizations is equivalent toone that randomly applies a discrete set of error operators.

The construction of the noncoherent space-time code may include forminga stabilizer group for this error set. The operators S₀=X⊗Z⊗X andS₁=X⊗X⊗Z satisfy the necessary commutation relations to form a set ofstabilizer generators, as summarized in Table 1.

TABLE 1 TABLE 1: SUMMARY OF COMMUTATION RELATIONS BETWEEN STABILIZER ANDERROR OPERATORS. C DENOTES COMMUTATION AND ADENOTES ANTI-COMMUTATIONCommutation Relationships S₀ S₁ E₀ C C E₁ C A E₂ A C E₃ A A

Because they commute, the stabilizer operators admit a partiallyintersecting +1 eigenspace, which has a two-dimensional basis spanned bythe vectors

v ₀=[1 0 0−1 0 1 1 0]^(T) and

v ₁=[0−1−1 0−1 0 0 1]^(T).

These vectors may be used to form a mapping that encodes two arbitrarycomplex numbers into a space-time codeword. Given a complex vectors=[s₁,s₂]^(T) from a general codebook, the vectorized space-timecodeword may be produced by applying an encoding operator C=[v₀, v₁]∈

^(8×2) yielding the following:

t=Cs

In some embodiments, it may be assumed that the symbol energy isnormalized, i.e., s*s=1. This assumption coupled with the definition ofC guarantees that t*t=4 which gives an average power of unity over thecoherence interval. The corresponding 2×4 code matrix for a codeword maybe represented with the inverse vectorization operator vec⁻¹:

⁸

^(2×4),

T = vec⁻¹(Cs), or $T = {\begin{bmatrix}s_{1} & {- s_{2}} & {- s_{2}} & s_{1} \\{- s_{2}} & {- s_{1}} & s_{1} & s_{2}\end{bmatrix}.}$

This code, which provides full diversity despite the noncoherentsetting, may be referred to as a generalized complex orthogonal design.

The symbol vector s may be viewed as an information carrying qubit statethat is to be preserved via the stabilizer encoding. Using theinterpretation of a qubit as a 1-dimensional subspace of

², it is assumed that symbol vectors s are drawn uniformly from aconstellation C. The constellations may be chosen as Grassmannian linepackings in

².

The detection rule described in more detail herein may motivate thechoice of the Grassmannian line packing for the qubit constellation.Consider, for example, the expectation

$B = {{\left\lbrack {{\sum\limits_{k = 0}^{3}\; {q_{k}q_{k}^{*}}}s} \right\rbrack} = {{4{ss}^{*}} + {2\sigma_{n}^{2}I_{2}}}}$

and the function R_(S)(ŝ) ŝBŝ*−ŝBŝ*, where s≠ŝ. This is the expectedvalue of the difference between computing the statistic on thetransmitted symbol as opposed to another, not transmitted symbol. It isexpected that the dominant error will occur when R_(S)(ŝ) is minimizedoveralls and s. Therefore, a constellation set with the maximal minimumR_(S)(ŝ) is desired. Since the transmit symbols are normalized, thedefinition of R_(S)(ŝ) indicates that for a N-point constellationencoding log₂(N) bits the set selected should be given by:

$\hat{C} = {\min\limits_{{C = {\{{{{s \in {\mathbb{C}}^{2}}{s^{*}s}} = 1}\}}},{{C} = N}}{\max \mspace{14mu} {{{{\hat{s}}^{*}s}}^{2}.}}}$

This indicates that the constellation should be chosen as a Grassmannianline packing. Using this approach, a choice of constellation may beshown to minimize a bound on the probability of a symbol detectionerror.

An example method 400 for encoding symbols using space-time encoding isillustrated by the flow diagram in FIG. 4, according to one embodiment.As illustrated at 402, the method may include determining a number, l,of stabilizer generators equal to twice the base-two logarithm of thenumber of transmitter antennas, or 2 log₂(N_(ant)).

At 404, the method may include defining a set of stabilizer generatorsincluding one stabilizer generator for each value of l from 0 to oneless than twice the base-two logarithm of the number of transmitterantennas, or 2 log₂(N_(ant))−1. In at least some embodiments, eachstabilizer generator S having an index value l of the plurality ofpredefined stabilizer generators may be defined as follows:

${S_{l} = {{X \otimes I_{2^{\lfloor\frac{}{2}\rfloor}} \otimes X \otimes I_{2^{k - 1}} \otimes X \otimes I_{2^{k - 1 - {\lfloor\frac{}{2}\rfloor}}}}\mspace{14mu} {when}\mspace{14mu} l\mspace{14mu} {is}\mspace{14mu} {even}}},{S_{l} = {{X \otimes I_{2^{\lfloor\frac{}{2}\rfloor}} \otimes Z \otimes I_{2^{k - 1}} \otimes Z \otimes I_{2^{k - 1 - {\lfloor\frac{}{2}\rfloor}}}}\mspace{14mu} {when}\mspace{14mu} l\mspace{14mu} {is}\mspace{14mu} {{odd}.}}}$

Here, l ranges from 0 to 2 log₂(N_(ant))−1, I represents an identitymatrix, and X and Z represent respective Pauli matrices defined asfollows:

${X = \begin{pmatrix}0 & 1 \\1 & 0\end{pmatrix}},{Z = {\begin{pmatrix}1 & 0 \\0 & {- 1}\end{pmatrix}.}}$

At 406, method 400 may include generating a projection matrix onto thevector space stabilized by the algebraic group generated by the set ofstabilizer generators. In at least some embodiments, the projectionmatrix may be defined as follows:

P ₀ =N _(ant) ⁻²Π_(q=0) ^(2 log) ² ^((N) ^(ant) ⁾⁻¹(I _(2N) _(ant) ₂ +S_(q)).

At 408, the method may include determining a space-time codespaceincluding the subset of available vectorized space-time codewords thatis invariant under the generated projection matrix. Once the space-timecodespace has been determined, it can be used to encode each of thesymbols without repeating the operations shown at 402 through 408.

At 410, method 400 may include encoding each symbol using a respectivevectorized space-time codeword in the determined space-time codespace ast=Cs. An example encoding, where

$\begin{pmatrix}\alpha \\\beta\end{pmatrix}\quad$

is a symbol mapped to a particular input block of bits and representedas an ordered pair of complex numbers is encoded using a chosen codewordC may be shown as:

$t = {{Cs} = {{{\frac{1}{\sqrt{2}}\begin{bmatrix}1 & 0 \\0 & 0 \\0 & 0 \\1 & 0 \\0 & 1 \\0 & 0 \\0 & 0 \\0 & 1\end{bmatrix}}\begin{pmatrix}\alpha \\\beta\end{pmatrix}} = {\frac{1}{\sqrt{2}}{\begin{pmatrix}\alpha \\0 \\0 \\\alpha \\\beta \\0 \\0 \\\beta\end{pmatrix}.}}}}$

In this setting of quantum-inspired classical coding, the ideas ofquantum measurement and syndrome decoding may be dispensed with in favorof the more familiar method of maximum likelihood (ML) inference. While,in at least some embodiments of the wireless communication systemsdescribed herein, the decoding process may be based on computing the MLdecoding rule, the decoding process may lend itself to a quantummechanical interpretation.

If it is assumed that the encoded symbol s is drawn uniformly from someconstellation C, the maximum a posteriori rule reduces to the canonicalML problem of finding s such that

$\hat{s} = {{\arg \mspace{14mu} {\max\limits_{s \in C}{f_{ys}\left( {sy} \right)}}} = {\arg \mspace{14mu} {\max\limits_{s \in C}{{f_{ys}\left( {ys} \right)}.}}}}$

Projection operators may be defined as follows:

P ₀=(I+S ₀)(I+S ₁)/4

P ₁=(I+S ₀)(I−S ₁)/4

P ₂=(I−S ₀)(I+S ₁)/4

P ₃=(I−S ₀)(I−S ₁)/4.

Note that, in this example, P₀ is the projector onto the codespace andP₀+P₁+P₂+P₃=I is the identity. The receiver computes the fourcorresponding projections of the received vector y onto the codespaceand the three error subspaces to obtain the following vectors:

P ₀ y=c ₀ t+P ₀ n

P ₁ y=c ₁ E ₁ t+P ₁ n

P ₂ y=c ₂ E ₂ t+P ₂ n

P ₃ y=c ₃ E ₃ t+P ₃ n,

where c₀, c₁, c₂, and c₃ are as defined above and based on the fact thatP_(k)E_(k)=E_(k)P₀. Since the projectors sum to identity, the vectorsabove are sufficient statistics for y. Recall that c=[c₀, c₁, c₂,c₃]^(T)˜

(0, I/2) and is independent of the noise.

The receiver may now carry out error correction on the projectedvectors. To do so, the receiver applies a unitary correction operatorE_(k) to each projection P_(k)y and obtains:

z ₀ =c ₀ t+n

z ₁ =c ₁ t+E ₁ P ₁ n

z ₂ =c ₂ t+E ₂ P ₂ n

z ₃ =c ₃ t+E ₃ P ₃ n.

Since the P_(k) are orthogonal projection operators, the projected andcorrected noise vectors, E_(k)P_(k)n, are mutually independent.Following from this, the commutation relationships and unitarity of thecorrection operators imply that the resulting noise vectors areidentically distributed with E_(k)P_(k)n˜

(0,σ_(n) ²P₀) for all k. Since t=Cs, projections of the z_(k) onto thecolumn space of C are sufficient to estimate s. Lettingn_(k)=C*E_(k)P_(k)n/(2√{square root over (2)}), and lettingĉ_(k)=√{square root over (2c_(k))} the receiver computes:

${q_{k} = {\frac{C^{*}z_{k}}{2\sqrt{2}} = {{{\hat{c}}_{k}s} + n_{k}}}},{{{for}\mspace{14mu} k} \in {\left\{ {0,1,2,3} \right\}.}}$

In this example, the n_(k) are independent and identically distributedwith n_(k)˜

(0,σ_(n) ² I/2. The scaled-identity covariance follows from the factthat P₀C=C, since the columns of C are by definition in the code, andthat C*C∝I.

The receive may now concatenate the q_(k) into the vector q=[q₀ ^(T), q₁^(T), q₂ ^(T), q₃ ^(T)]^(T) and reformulate the maximum likelihoodproblem as:

$\hat{s} = {\arg \mspace{14mu} {\max\limits_{s}\mspace{14mu} {f_{qs}\left( {qs} \right)}}}$

Given the transmit symbol s, q is a Gaussian random vector. Here, w maybe defined as w=[ĉ^(T), n₀ ^(T), n₁ ^(T), n₂ ^(T), n₃ ^(T)]^(T) so thatw˜

(0, Σ), wherein

$\Sigma = {\begin{bmatrix}I_{4} & 0_{4 \times 8} \\0_{8 \times 4} & {\frac{\sigma_{n}^{2}}{2}I_{8}}\end{bmatrix}.}$

Defining the matrix M∈

^(8×12) via

M=[(I ₄ ⊗s)I ₈]

yields

q=Mw.

Thus, q˜

(0, Q), where Q=MΣM*. It can be shown that

$Q = {I_{4} \otimes {\left( {{ss}^{*} + {\frac{\sigma_{n}^{2}}{2}I_{2 \times 2}}} \right).}}$

It has been discovered that the second definition of Q may be useful insimplifying the likelihood function. Assuming that Q is invertible, thelikelihood function can be written as

${f_{q|s}\left( q \middle| s \right)} = {\frac{\exp \left( {{- q^{*}}Q^{- 1}q} \right)}{\pi^{8}{\det (Q)}}.}$

Using the property of determinants of Kronecker products yields

${\det (Q)} = {{\det \left( {{ss}^{*} + {\frac{\sigma_{n}^{2}}{2}I_{2 \times 2}}} \right)}^{4}.}$

Since, by assumption

${{s^{*}s} = 1},{{\det (Q)} = \left\lbrack {\left( {1 + \frac{\sigma_{n}^{2}}{2}} \right)\frac{\sigma_{n}^{2}}{2}} \right\rbrack^{4}},$

which is constant in s. Furthermore, using the Kronecker productdefinition, it is clear that

$Q^{- 1} = {I_{4x4} \otimes {\left( {{ss^{*}} + {\frac{\sigma_{n}^{2}}{2}I_{2x2}}} \right)^{- 1}.}}$

Designating

$U_{s} = \left( {{ss^{*}} + {\frac{\sigma_{n}^{2}}{2}I_{2x2}}} \right)^{- 1}$

and calculating the inverse explicitly yields

${U_{s} = {\frac{1}{\frac{\sigma_{n}^{2}}{2}\left( {1 + \frac{\sigma_{n}^{2}}{2}} \right)}\begin{bmatrix}\left| s_{2} \middle| {}_{2}{+ \frac{\sigma_{n}^{2}}{2}} \right. & {{- s_{1}}s_{2}^{*}} \\{{- s_{2}}s_{1}^{*}} & \left| s_{1} \middle| {}_{2}{+ \frac{\sigma_{n}^{2}}{2}} \right.\end{bmatrix}}},$

which, through substitution, yields an explicit decision rule, asfollows:

$\hat{s} = {\underset{s \in C}{argmin}{q^{*}\left( {I_{4 \times 4} \otimes U_{s}} \right)}{q.}}$

This may be further simplified by noting that, since s is normalized,

$U_{s} \propto {{\frac{\sigma^{2}}{2}I_{2}} + {\left( {I_{2} - {ss^{*}}} \right).}}$

Thus, the decision rule can be written as:

$\hat{s} = {{\underset{s \in C}{argmax}{\sum\limits_{k = 0}^{3}\; {q_{k}^{*}{ss}^{*}q_{k}}}} = {\underset{s \in C}{argmax}s^{*}{\sum\limits_{k = 0}^{3}\; {\left( {q_{k}q_{k}^{*}} \right){s.}}}}}$

This form of the decoding rule lends itself to a quantum mechanicalinterpretation. For example, {circumflex over (q)}_(k){circumflex over(q)}_(k)*=q_(k)q_(k)*/tr(q_(k)q_(k)*) may be interpreted as normalizeddensity operators. The mixed state, Ψ, may be formed from drawing thestates {circumflex over (q)}_(k){circumflex over (q)}_(k)* withrespective probabilities as follows:

$p_{k} = {\frac{{tr}\left( {q_{k}q_{k}^{*}} \right)}{\sum_{i = 0}^{3}{{tr}\left( {q_{i}q_{i}^{*}} \right)}}.}$

This yields the density matrix:

$\psi = \frac{\Sigma_{i = 0}^{3}\left( {q_{i}q_{i}^{*}} \right)}{\Sigma_{i = 0}^{3}t{r\left( {q_{i}q_{i}^{*}} \right)}}$

which is the same matrix that appears on the right-hand side of thedecision rule shown above, up to a positive scale factor. Thus, usingthe definition of fidelity described earlier, it can be seen that the MLdetection rule consists of finding the input state that maximizes thefidelity with respect to ψ, or, more explicitly,

$\hat{s} = {\underset{s \in C}{argmax}{{F\left( {\psi,{ss}^{*}} \right)}.}}$

Note that maximizing the fidelity is the same as maximizing its square.

An example method 500 for space-time decoding is illustrated by the flowdiagram in FIG. 5, according to one embodiment. As shown at 502, themethod may include receiving, collectively at multiple antennas of areceiver, electrical signals representing a transmitted input block anddefined by a vectorized space-time codeword produced using a space-timeencoding based on predefined stabilizer generators. More specifically,the m receiving antennas receive m different electrical signals at eachtime instant. The received signals are downconverted, after whichsignals received during a coherence interval, e.g., TC=2m time instants,and then concatenated into a vector y such that the m signals receivedat a first time instant are followed by the m signals received at asecond time instant, and so on up to the 2m^(th) time instant.

At 504, the method may include computing one vector projection per erroroperator for a total number of projections of y onto respectivesubspaces is equal to the square of the number of antennas, or m². Forexample, the projectors may be computed as elements of a projectionmatrix of the form:

=2^(−2k)Π_(q=0) ^(2k-1)

which defines a projection onto a 2D subspace. The resulting projectionsmay be defined as

=P_(i)y. Note that C is a matrix whose columns are a basis for therange, or column space, of P₀, where C∈

^(2m) ² ⁺².

At 506, method 500 may include rotating each projection back into thecodespace by inverting the error using one error operator perprojection. The result may be defined as

=E_(i)

. Examples of the projections and rotations described herein areillustrated in FIGS. 6A and 6B and described below.

At 508, method 500 may include computing scalar projections onto 2Dcodespace, where the number of scalar projections is equal to the squareof the number of antennas, or m².

At 510, the method may include correlating the 2D projections withn-tuple complex vectors representing symbols to which an input block mayhave been mapped by the transmitter. For example, the correlations of

with codewords may be computed via

=C*

.

At 512, method 500 may include identifying a symbol having the highestcorrelation with the 2D projections, by computing the following:

$\hat{s} = {\underset{s \in C}{argmax}{\sum_{i = 0}^{m^{2} - 1}{s^{*}w_{i}w_{i}^{*}{s.}}}}$

This produces a symbol chosen from the set of possible input symbolsthat is the maximum likelihood estimate for the symbol produced at thetransmitter.

At 514, the method may include recovering the input block from theidentified symbol. The input block may be recovered from the set of2^(q) symbols in the constellation used by the symbol mapper of thetransmitter, where q is the number of bits in each input block, andthere is a 1-to-1 mapping between possible input blocks and availablesymbols, as described above.

For the projection matrices described herein, P₀ is a projector onto theintersection of the +1 eigenspaces of both of the generators, S₀ and S₁,P₁ is a projector onto the intersection of the +1 eigenspace of S₀ andthe −1 eigenspace of S₁, P₂ is the converse, and P₃ is the projectiononto the intersection of the −1 eigenspaces of both of the generators.

The equations below illustrate what happens when the vectorized receivedsignal Ht is multiplied by one of the projectors. Each projector,

, annihilates all of the terms except the one involving error i. This isbecause the labeling has been chosen carefully. Thus, if the receivedsignal is further pre-multiplied by the inverse of the error operator,which is its Hermitian conjugate, the space-time codeword may berecovered up to a random complex (Gaussian) constant.

H t=

Ht=

FIGS. 6A and 6B illustrate an example application of a projection matrixand error operators, in accordance with some embodiments. In thesefigures, each of the bulbs represents a subspace of the codespace C(S).Due to the stabilizer formalism, these subspaces are orthogonal. Eacherror operator,

, either rotates the codespace into one of the three other subspaces ordoes not do anything at all. These subspaces may be referred to as“error subspaces.” In general, the received signal has energy in all thesubspaces, since the channel is a linear combination of all of the erroroperators. Multiplying the received signal by one of those subspaceprojectors eliminates all of the signal energy outside of the particularsubspace. Further multiplication by the error operator rotates thatsubspace back into the codespace.

FIGS. 6A and 6B illustrate example visualizations of the mappingsbetween error operators and subspaces described by projection operators.More specifically, FIG. 6A illustrates the application of each erroroperator, E_(i) to a particular subspace, which is defined by theprojector P₀ (602). For example, E₁ (614) rotates the subspace definedby P₀ (602) to the subspace defined by P₁(604), E₂ (616) rotates thesubspace defined by P₀ (602) to the subspace defined by P₂ (608), and E₃(618) rotates the subspace defined by P₀ (602) to the subspace definedby P₃ (606), but E₀ (612) has no effect on the subspace defined by P₀(602).

FIG. 6B illustrates the application of the inversion of each erroroperator, which is its Hermitian conjugate, to particular subspaces. Forexample, E₁*(624) rotates the subspace defined by P₁ (604) back to thesubspace defined by P₀ (602), E₂*(626) rotates the subspace defined byP₂ (608) back to the subspace defined by P₀, (602), and E₃*(628) rotatesthe subspace defined by P₃ (606) back to the subspace defined by P₀(602), but E₀*(622) has no effect.

In the example wireless communication system described above, after theprojection and “error correction” steps, there will be four vectors,each of which should be proportional (up to a Gaussian scalar) to thetransmitted space-time codeword. In reality, there will also be additivenoise. In at least some embodiments, the correlation techniquesdescribed above in reference to FIG. 5 may be used to estimate thetransmitted symbol based on the recovered codewords.

Numerical simulations have been carried out to demonstrate theperformance of the noncoherent space-time code presented herein. Thesesimulations included Grassmannian line packings of size N=4 and N=8 in

² with a Rayleigh fading environment.

The simulations compared the stabilizer-based, noncoherent space-timecode constructions described herein with a coherent scheme based on theAlamouti code (for 2×2 systems) at spectral efficiency rates of r=½ andr=1 bits/channel use. Channel estimation was first performed bytransmitting the symbols [1,1]^(T)/√{square root over (2)} and [1,−1]^(T)/√{square root over (2)} solving for an estimate of H at thereceiver. Subsequently, the Alamouti scheme was used to transmit onespace-time symbol s∈

² over the remaining two channel uses in the coherence interval. Encodedin s were two binary phase-shift keying (BPSK) symbols for the rate r=½approach and two quadrature phase-shift keying (QPSK) symbols for therate r=1 approach.

Similarly, the simulations compared the stabilizer-based, noncoherentspace-time code constructions described herein with an approach usingdifferential unitary group codes. At both r=½ and r=1, the first twochannel uses were used for the 2×2 reference matrix, and no informationwas transmitted. With the next two channel uses, a single differentiallyencoded 2×2 matrix drawn from an appropriately sized constellation wastransmitted. For r=½, this constellation is a group code over the QPSKconstellation; specifically, encoding over the 2×2 Pauli group elements.For r=1, this constellation is a dicylic group code generated over16-PSK. The transmit symbols in both sets of comparisons wereappropriately normalized so that the transmit power was constant overthe four transmissions.

Graph 700 in FIG. 7 plots bit error probability vs. SNR for thedisclosed space-time encoding techniques as well as other encodingtechniques. More specifically, the disclosed space-time encodingtechniques were applied to a noncoherent MIMO channel of a 2×2 antennasystem with T=4, m1=m2=2, where symbols were drawn from Grassmannianline packings (with 4 and 8 symbols). The disclosed space-time codeswere compared to two other codes with approximately the same data rates,including the Alamouti coherent space-time code described above and thedifferential noncoherent code described above. Using an exact comparisonfor a rate 1/2 code, the disclosed space-time codes outperformed theAlamouti coherent space-time code slightly, and significantlyoutperformed the differential code. At other rates, such as a rate 3/4code and a rate 1 code, results were mixed, with the disclosedspace-time codes significantly outperforming a rate 1 differential code,and slightly outperforming an Alamouti code at rate 1.

A useful bound on the probability of symbol error has been computed asdescribed below. In computing the bound, it was assumed that aparticular symbol s₀ was transmitted. The approach included consideringwhether a sufficient statistic for estimating s (e.g., the sum of outerproducts of rotated projections) is closer to s₁, is closer to s₂, or iscloser to s₃. This is the probability of a symbol error given that s₀was transmitted:

S ₀ →S ₁ =F(s ₀ ,{circumflex over (Q)})≤F(s _(i) ,{circumflex over (Q)})

A typical wireless approach of using a union bound over those events isshown below as:

P _(e|0)=

(

_(≠0) s ₀→

0), where

(

_(≠0) s ₀→

0)≤

₌₀

(s ₀→

0).

The union bound may then be relaxed by replacing each term s₀→

with the maximum of those terms, as follows:

${P_{e|}0} \leq {\left( \left| C \middle| {- 1} \right. \right){\max\limits_{i}{{{\mathbb{P}}\left( {s_{0}->{s_{i|}0}} \right)}.}}}$

Assuming all input symbols are equally likely to be transmitted, theprobability of error may be bound by maximizing the aforementioned (andanalogous) bounds over all possible input symbols, as follows:

$P_{e} \leq {\max\limits_{}P_{e|}} \leq {\left( \left| C \middle| {- 1} \right. \right){\max\limits_{i}{{{\mathbb{P}}\left( {S_{}->{S_{i|}}} \right)}.}}}$

All that is needed for this bound is the maximum pairwise errorprobability, shown as the last term in the equation above. It has beendiscovered that all of the pairwise error probabilities are easy tobound using the Chernoff bound, which is a geometric bound on thecumulative distribution function of sums of independent and identicallydistributed random variables. The bound on the probability of symbolerror thus becomes:

${P_{e} \leq {\left( {{C} - 1} \right)\left( {1 - \frac{1 - {{\hat{\delta}}2}}{\left( \frac{1}{SNR} \right)^{2} + \frac{2}{SNR} + \left( {1 - {\hat{\delta}}^{2}} \right)}} \right)N_{ant}^{2}}},$

where P_(e)→0 as SNR→∞. In other words, the error probability goes to 0as the SNR goes to infinity. Furthermore, the bound depends on the normof S, shown below:

${\hat{\delta}}^{2} = {\underset{m \neq n}{\max\limits_{S_{m},{S_{n} \in C}}}{{s_{m}^{*}s_{n}}}^{2}}$(δ̂² ∈ [0, 1])

This essentially represents the maximum absolute correlation betweenelements of the qubit symbol constellation, which increases withincreasing correlation. Thus, the choice of a Grassmannian line packingminimizes an upper bound on the probability of error.

Results of simulations shown in FIGS. 8A and 8B illustrate that theexpected bound is, in fact, a bound. For example, graph 800 in FIG. 8Aplots signal-to-noise ratio (SNR) vs. symbol error rate fortransmitters/receivers with different numbers of antennas (in this case,2, 4, and 8 antennas) with different data rates, with and withoutbounding. Similarly, graph 801 in FIG. 8B plots SNR vs. symbol errorrate for transmitters/receivers with different numbers of antennas (inthis case, 16, 32, and 64 antennas) with different data rates, with andwithout bounding.

While several examples of noncoherent space-time encoding are describedherein in terms of wireless communication systems that include atransmitter with two antennas and a receiver with two antennas, in otherembodiments, the disclosed techniques may be applied in systems thatinclude more than two antennas at the transmitter and/or at thereceiver, or different numbers of transmitters at the transmitter thanat the receiver. In some embodiments, these techniques may be extendedto “square” antenna configurations with a dyadic number of antennas(i.e. N_(ant)=N_(TX)=N_(RX)−2^(k)). In such embodiments, the number oferror operators would be (N_(ant))² and the number of stabilizergenerators would be 4^(k).

In general, a space-time block code model with N transmitter antennas, Mreceiver antennas, and an integer coherence length T may have the form

$\begin{matrix}{Y = {{HX}.}} & \; \\{where} & \; \\{\begin{pmatrix}Y_{1,1} & \ldots & Y_{1,T} \\\vdots & \ddots & \vdots \\Y_{M,1} & \ldots & Y_{M,T}\end{pmatrix} = {\begin{pmatrix}H_{1,1} & \ldots & H_{1,N} \\\vdots & \ddots & \vdots \\H_{M,1} & \ldots & H_{M,N}\end{pmatrix}{\begin{pmatrix}X_{1,1} & \ldots & Y_{1,T} \\\vdots & \ddots & \vdots \\X_{N,1} & \ldots & Y_{N,T}\end{pmatrix}.}}} & \;\end{matrix}$

This expression may be vectorized by stacking the columns of Y and X,resulting in:

${\begin{bmatrix}\begin{pmatrix}Y_{1,1} \\\vdots \\Y_{1,T}\end{pmatrix} \\\vdots \\\begin{pmatrix}Y_{M,1} \\\vdots \\Y_{M,T}\end{pmatrix}\end{bmatrix} = {\begin{bmatrix}H & 0 & \ldots & 0 \\0 & H & 0 & \ldots \\\vdots & \; & \ddots & \vdots \\0 & \ldots & 0 & H\end{bmatrix}\begin{bmatrix}\begin{pmatrix}X_{1,1} \\\vdots \\X_{1,T}\end{pmatrix} \\\vdots \\\begin{pmatrix}X_{N,1} \\\vdots \\X_{N,T}\end{pmatrix}\end{bmatrix}}},$

which may be written more compactly as

y=(I _(T) ⊗H)x.

Note that H can be decomposed as a superposition of matrices B₀, . . . ,B_(MN-1) that form an orthonormal basis with respect to theHilbert-Schmidt inner product, so that

$H = {\sum\limits_{i = 0}^{{MN} - 1}{h_{i}B_{i}}}$

where h_(i) are complex random variables assumed to be such that

E[

h _(j)*]=

and

Tr[

B _(j)]=

.

Therefore, this classical channel may be construed as a quantum channelthat transforms a pure state xx* into the mixed state ρ=E[yy*|x], since

$\begin{matrix}{\rho = {{E\left\lbrack {\left( {I_{T} \otimes H} \right){{xx}^{*}\left( {I_{T} \otimes H} \right)}^{*}} \middle| x \right\rbrack} = {{\sum_{i,j}{{E\left\lbrack {h_{i}h_{j}^{*}} \middle| x \right\rbrack}\left( {I_{T} \otimes B_{i}} \right){{xx}^{*}\left( {I_{T} \otimes B_{j}^{*}} \right)}}} = {{\sum\limits_{i = 0}^{{MN} - 1}{\left( {I_{T} \otimes B_{i}} \right){{xx}^{*}\left( {I_{T} \otimes B_{i}} \right)}^{*}}} = {\sum\limits_{i = 0}^{{MN} - 1}{E_{i}{xx}^{*}E_{i}^{*}}}}}}} & \; \\{\mspace{79mu} {where}} & \; \\{\mspace{79mu} {E_{i} = {I_{T} \otimes B_{i}}}} & \;\end{matrix}$

are the MT×NT Kraus error matrices.

It may be supposed that the errors are invertible in the sense thatthere exists an MT×NT left inverse

such that

=I _(NT).

In the special case in which

is a member of the Pauli group,

=E*. More generally, if

has linearly independent columns (and hence B*

is invertible), then

=(E*

)⁻¹E*, the pseudoinverse of

. It may be further supposed that the errors are orthogonal in the sensethat

F _(j)

=

I _(NT).

For encoding and error recovery, x may be considered an encoding of asymbol s∈C^(d), where d≥2 and s is such that ∥s∥=1. Hence, ss* may beconstrued as a pure quantum state. The transmitted vector x may beobtained from s via application of a NT×d encoding matrix C such that

x=Cs.

It may be supposed that C is such that application of C* to x recoverss. In other words,

C*C=I _(d).

Given the code matrix C and an error matrix E_(j), a projection matrixP_(j) onto the error subspace may be constructed as follows:

P _(j) =E _(j) CC*F _(j)

Here, each P_(j) is indeed a projection matrix, since

P_(j)² = E_(j)CC^(*)(F_(j)E_(j))CC^(*)F_(j) = E_(j)C(C^(*)C)C^(*)F_(j) = E_(j)CC^(*)F_(j) = P_(j).

Here, note that the projection matrix leaves the corresponding errorsubspace invariant. Specifically,

P _(j)

C=

E _(j) C,

since

P _(j)

C=E _(j) CC*F _(j)

C=E _(j) CC*(

I _(NT))C=

E _(j) C.

An error recovery process may now be defined as follows. Given y, applyC*F_(j)P_(j) to yield

${C^{*}F_{j}P_{j}y} = {{C^{*}F_{j}{P_{j}\left( {\sum\limits_{i = 0}^{{MN} - 1}{h_{i}E_{i}}} \right)}{Cs}} = {{\sum\limits_{i = 0}^{{MN} - 1}{h_{i}C^{*}F_{j}P_{j}E_{i}{Cs}}} = {{h_{j}C^{*}F_{j}E_{j}{Cs}} = {h_{j}{s.}}}}}$

Each projection of y yields a scalar multiple of s. Assuming these arenot all zero (which is true almost surely), the value of s may berecovered uniquely. Note that this procedure is only valid for d≥2. Inparticular, it would not work for QAM symbols (for which d=1).

In one example, N=1, M=2, and T=2. Let B₀=[1 0]^(T) and B₁=[0 1]^(T), soH=[h₀h₁]^(T). The error matrices are, then,

${E_{0} = \begin{bmatrix}1 & 0 \\0 & 0 \\0 & 1 \\0 & 0\end{bmatrix}},{E_{1} = \begin{bmatrix}0 & 0 \\1 & 0 \\0 & 0 \\0 & 1\end{bmatrix}}$

and their left inverses are F₀=E₀ ^(T) and F₁=E₁ ^(T). Note thatF_(i)E_(j)=δ_(ij)I₂. In particular, take d=2 and define C=I₂ (repetitioncode). Then P_(j)=E_(j)F_(j).

FIG. 9 illustrates an example computing system 900 for space-timeencoding in a wireless communication system, according to oneembodiment. In some embodiments, computing system 900 may be, or may bea component of, a transmitter or receiver in a wireless communicationsystem. In other embodiments, computing system 900 may be a stand-alonesystem that provides information to a transmitter or receiver toconfigure it for space-time encoding and/or decoding. In this exampleembodiment, computing system 900 includes one or more processors 910.Each of processors 910 may include circuitry or logic to interpret orexecute program instructions and/or to process data. For example, eachprocessor 910 may include a microprocessor, microcontroller, digitalsignal processor (DSP), graphics processor, field-programmable gatearray (FPGA), or application specific integrated circuit (ASIC). In someembodiments, processors 910 may interpret and/or execute programinstructions and/or process data stored locally in memory subsystem 920or remotely (not shown).

Processors 910 may implement any instruction set architecture (ISA), indifferent embodiments. In some embodiments, all of the processors 910may implement the same ISA. In other embodiments, two or more ofprocessors 910 may implement different ISAs. Processors 910 are coupledto a memory subsystem 920, a network interface 955, and an input/outputsubsystem 950 via a system interface 915. System interface 915 mayimplement any of a variety of suitable bus architectures and protocolsincluding, but not limited to, a Micro Channel Architecture (MCA) bus,Industry Standard Architecture (ISA) bus, Enhanced ISA (EISA) bus,Peripheral Component Interconnect (PCI) bus, PCI-Express bus, or aHyperTransport (HT) bus.

In some embodiments, memory subsystem 920 may include random accessmemory (RAM), read-only memory (ROM), electrically erasable programmableread-only memory (EEPROM), flash memory, magnetic storage, opto-magneticstorage, and/or any other type of volatile or non-volatile memory. Insome embodiments, memory subsystem 920 may include computer-readablemedia, e.g., a hard disk drive, floppy disk drive, CD-ROM, and/or othertype of rotating storage media, and/or another type of solid-statestorage media. In the example embodiment illustrated in FIG. 9, memorysubsystem 920 includes program instructions 930, including programinstructions that when executed by one or more of the processors 910 mayimplement all or a portion of one or more of the methods describedherein for space-time encoding in a wireless communication system, suchas method 300 illustrated in FIG. 3, method 400 illustrated in FIG. 4,and/or method 500 illustrated in FIG. 5. For example, programinstructions 930 include symbol mapper 932, space-time encoder 934,symbol de-mapper 936, and/or space-time decoder 938. In someembodiments, these components may implement functionality of atransmitter and/or a receiver in a wireless communication system, suchas those described herein. In the example embodiment illustrated in FIG.9, storage 940 includes storage for symbol mappings (shown as 942)and/or storage for space-time codewords (shown as 944). Storage 940 mayalso store other data used in performing the methods described herein,such as definitions of stabilizer generators or corresponding algebraicgroups, definitions of projection matrices, Pauli group matrices, erroroperators, or characteristics of a transmitter or receiver, includingthe number of transmitting or receiving antennas (not shown in FIG. 9).Storage 940 may also store data used in implementing functionality ofcomputing system 900 other than space-time encoding.

In the example embodiment illustrated in FIG. 9, input/output subsystem950 may implement any of a variety of digital and/or analogcommunication interfaces, graphics interfaces, video interfaces, userinput interfaces, and/or peripheral interfaces for communicativelycoupling input/output devices or other remote devices to the componentsof computing system 900. Input/output subsystem 950 may generate signalsto be provided to one or more input devices 970. Input/output subsystem950 may also generate signals to be provided to a display device 960.For example, display device 960 may receive signals encoding data fordisplay of a graphical user interface (GUI) or command line interfacefor interacting with various components of a computing system 900 toinitiate the performance of method 300 illustrated in FIG. 3, method 400illustrated in FIG. 4, or method 500 illustrated in FIG. 5, or todisplay results.

In the example embodiment illustrated in FIG. 9, network interface 955may implement any of a variety of digital and/or analog networkinterfaces, in accordance with any suitable network protocols, to allowcomputing system 900 to interact with various transmitters and/orreceivers 965. Alternatively, computing system 900 may interact withtransmitters and/or receivers 965 using a digital and/or analogcommunication interface implemented by input/output subsystem 950. Forexample, computing system 900 may, after generating symbol mappings 942or space-time codewords 944, provide the symbol mappings 942 orspace-time codewords 944 to a transmitter 965 via network interface 955or input/output subsystem 950 to configure the transmitter to performspace-time encoding when transmitting information in a wirelesscommunication system, as described herein. Similarly, computing system900 may provide symbol mappings, definitions of projection matrices, orerror operators to a receiver 965 to configure the receiver to performspace-time decoding when receiving transmitted information that wasencoded using space-time encoding, as described herein.

As noted above, in some embodiments, computing system 900 may be, or maybe a component of, a transmitter or receiver in a wireless communicationsystem. In some such embodiments, computing system 900 may includecircuitry to collectively transmit, by a plurality of antennas of thetransmitter to a plurality of antennas of a receiver over a wirelesstransmission channel, electrical signals defined by the vectorizedspace-time codewords described herein (not shown in FIG. 9). In somesuch embodiments, computing system 900 may include circuitry tocollectively receive, by a plurality of antennas of the receiver from aplurality of antennas of a transmitter over a wireless transmissionchannel, electrical signals defined by the vectorized space-timecodewords described herein (not shown in FIG. 9).

Although only exemplary embodiments of the present disclosure arespecifically described above, it will be appreciated that modificationsand variations of these examples are possible without departing from thespirit and intended scope of the disclosure.

The above disclosed subject matter is to be considered illustrative, andnot restrictive, and the appended claims are intended to cover all suchmodifications, enhancements, and other embodiments which fall within thetrue spirit and scope of the present disclosure. Thus, to the maximumextent allowed by the law, the scope of the present disclosure is to bedetermined by the broadest permissible interpretation of the followingclaims and their equivalents and shall not be restricted or limited bythe foregoing detailed description.

What is claimed is:
 1. A transmitter for wireless communication,comprising: a first plurality of antennas; a symbol mapper configured tomap an input block of an incoming binary bit stream representinginformation to be transmitted to a symbol representing an orderedplurality of complex numbers, the input block comprising a plurality ofbinary bits; a space-time encoder configured to apply an encodingoperator to the symbol to produce a vectorized space-time codeworddefining electrical signals to be transmitted by the transmitter, theencoding operator being dependent on a plurality of predefinedstabilizer generators; and circuitry to collectively transmit, by thefirst plurality of antennas to a second plurality of antennas of areceiver over a wireless transmission channel, the electrical signalsdefined by the vectorized space-time codeword.
 2. The transmitter ofclaim 1, wherein the wireless transmission channel is a noncoherentmultiple-input multiple-output (MIMO) channel characterized by anunknown channel matrix and is assumed to have coherence over apredefined time period and additive Gaussian noise.
 3. The transmitterof claim 1, wherein the number of antennas in the first plurality ofantennas and the number of antennas in the second plurality of antennasare the same.
 4. The transmitter of claim 1, wherein the number ofantennas in each of the first plurality of antennas and the secondplurality of antennas is equal to a power of two.
 5. The transmitter ofclaim 1, wherein the symbol represents an ordered pair of complexnumbers.
 6. The transmitter of claim 1, wherein: each stabilizergenerator S having an index value l of the plurality of predefinedstabilizer generators is defined as follows:${S_{l} = {{X \otimes I_{2^{\lfloor\frac{}{2}\rfloor}} \otimes X \otimes I_{2^{k - 1}} \otimes X \otimes I_{2^{k - 1 - {\lfloor\frac{}{2}\rfloor}}}}\mspace{14mu} {when}\mspace{14mu} l\mspace{14mu} {is}\mspace{14mu} {even}}},{{S_{l} = {{X \otimes I_{2^{\lfloor\frac{}{2}\rfloor}} \otimes Z \otimes I_{2^{k - 1}} \otimes Z \otimes I_{2^{k - 1 - {\lfloor\frac{}{2}\rfloor}}}}\mspace{14mu} {when}\mspace{14mu} l\mspace{14mu} {is}\mspace{14mu} {odd}}};}$l ranges from 0 to one less than twice the base-two logarithm of thenumber of antennas in the first plurality of antennas; I represents anidentity matrix; and X and Z represent respective Pauli matrices definedas follows: ${X = \begin{pmatrix}0 & 1 \\1 & 0\end{pmatrix}},{Z = {\begin{pmatrix}1 & 0 \\0 & {- 1}\end{pmatrix}.}}$
 7. The transmitter of claim 1, wherein the symbolmapper is dependent on a target data rate for transmission of theinformation.
 8. A method for transmitting information in a wirelesscommunication system, comprising: receiving an incoming binary bitstream representing the information; and for each input block of theincoming binary bit stream comprising a predefined number of binarybits: mapping the input block to a respective symbol representing anordered plurality of complex numbers; applying an encoding operator tothe respective symbol to produce a respective vectorized space-timecodeword defining electrical signals to be transmitted, the encodingoperator being dependent on a plurality of predefined stabilizergenerators; and transmitting, collectively and by a first plurality ofantennas of a transmitter to a second plurality of antennas of areceiver over a wireless transmission channel, the electrical signalsdefined by the vectorized space-time codeword.
 9. The method of claim 8,wherein the wireless transmission channel is a noncoherentmultiple-input multiple-output (MIMO) channel characterized by anunknown channel matrix and is assumed to have coherence over apredefined time period and additive Gaussian noise.
 10. The method ofclaim 8, wherein the number of antennas in the first plurality ofantennas and the number of antennas in the second plurality of antennasare the same.
 11. The method of claim 8, wherein the number of antennasin each of the first plurality of antennas and the second plurality ofantennas is equal to a power of two.
 12. The method of claim 8, whereinthe symbol represents an ordered pair of complex numbers.
 13. The methodof claim 8, wherein: each stabilizer generator S having an index value lof the plurality of predefined stabilizer generators is defined asfollows:${S_{l} = {{X \otimes I_{2^{\lfloor\frac{}{2}\rfloor}} \otimes X \otimes I_{2^{k - 1}} \otimes X \otimes I_{2^{k - 1 - {\lfloor\frac{}{2}\rfloor}}}}\mspace{14mu} {when}\mspace{14mu} l\mspace{14mu} {is}\mspace{14mu} {even}}},{{S_{l} = {{X \otimes I_{2^{\lfloor\frac{}{2}\rfloor}} \otimes Z \otimes I_{2^{k - 1}} \otimes Z \otimes I_{2^{k - 1 - {\lfloor\frac{}{2}\rfloor}}}}\mspace{14mu} {when}\mspace{14mu} l\mspace{14mu} {is}\mspace{14mu} {odd}}};}$l ranges from 0 to one less than twice the base-two logarithm of thenumber of antennas in the first plurality of antennas; I represents anidentity matrix; and X and Z represent respective Pauli matrices definedas follows: ${X = \begin{pmatrix}0 & 1 \\1 & 0\end{pmatrix}},{Z = {\begin{pmatrix}1 & 0 \\0 & {- 1}\end{pmatrix}.}}$
 14. The method of claim 8, wherein the symbol mapperis dependent on a target data rate for transmission of the information.15. A system for multiple-input multiple-output (MIMO) transmission,comprising: a transmitter comprising a first plurality of antennas; anda receiver comprising a second plurality of antennas; wherein thetransmitter further comprises: a symbol mapper configured to map aninput block of an incoming binary bit stream representing information tobe transmitted to a symbol representing an ordered pair of complexnumbers, the input block comprising a plurality of binary bits; aspace-time encoder configured to apply an encoding operator to thesymbol to produce a vectorized space-time codeword defining electricalsignals to be transmitted by the transmitter, the encoding operatorbeing dependent on a plurality of predefined stabilizer generators; andcircuitry to collectively transmit, by the first plurality of antennasof the transmitter to the second plurality of antennas of the receiverover a wireless transmission channel, the electrical signals defined bythe vectorized space-time codeword.
 16. The system of claim 15, whereinthe receiver comprises: a space-time decoder configured to recover thesymbol from the electrical signals transmitted by the transmitter; and asymbol de-mapper configured to recover the input block from the symbol.17. The system of claim 15, wherein the wireless transmission channel isa MIMO channel characterized by an unknown channel matrix and is assumedto have coherence over a predefined time period and additive Gaussiannoise.
 18. The system of claim 15, wherein the number of antennas ineach of the first plurality of antennas and the second plurality ofantennas is two.
 19. The system of claim 15, wherein: each stabilizergenerator S having an index value l of the plurality of predefinedstabilizer generators is defined as follows:${S_{l} = {{X \otimes I_{2^{\lfloor\frac{}{2}\rfloor}} \otimes X \otimes I_{2^{k - 1}} \otimes X \otimes I_{2^{k - 1 - {\lfloor\frac{}{2}\rfloor}}}}\mspace{14mu} {when}\mspace{14mu} l\mspace{14mu} {is}\mspace{14mu} {even}}},{{S_{l} = {{X \otimes I_{2^{\lfloor\frac{}{2}\rfloor}} \otimes Z \otimes I_{2^{k - 1}} \otimes Z \otimes I_{2^{k - 1 - {\lfloor\frac{}{2}\rfloor}}}}\mspace{14mu} {when}\mspace{14mu} l\mspace{14mu} {is}\mspace{14mu} {odd}}};}$l ranges from 0 to one less than twice the base-two logarithm of thenumber of antennas in the first plurality of antennas; I represents anidentity matrix; and X and Z represent respective Pauli matrices definedas follows: ${X = \begin{pmatrix}0 & 1 \\1 & 0\end{pmatrix}},{Z = {\begin{pmatrix}1 & 0 \\0 & {- 1}\end{pmatrix}.}}$
 20. The system of claim 15, wherein the symbol mapperis dependent on a target data rate for transmission of the information.